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Chapter Introduction
NAG Toolbox

NAG Toolbox: nag_lapack_dspsv (f07pa)

 Contents

    1  Purpose
    2  Syntax
    7  Accuracy
    9  Example

Purpose

nag_lapack_dspsv (f07pa) computes the solution to a real system of linear equations
AX=B ,  
where A is an n by n symmetric matrix stored in packed format and X and B are n by r matrices.

Syntax

[ap, ipiv, b, info] = f07pa(uplo, ap, b, 'n', n, 'nrhs_p', nrhs_p)
[ap, ipiv, b, info] = nag_lapack_dspsv(uplo, ap, b, 'n', n, 'nrhs_p', nrhs_p)

Description

nag_lapack_dspsv (f07pa) uses the diagonal pivoting method to factor A as A=UDUT if uplo='U' or A=LDLT if uplo='L', where U (or L) is a product of permutation and unit upper (lower) triangular matrices, D is symmetric and block diagonal with 1 by 1 and 2 by 2 diagonal blocks. The factored form of A is then used to solve the system of equations AX=B.

References

Anderson E, Bai Z, Bischof C, Blackford S, Demmel J, Dongarra J J, Du Croz J J, Greenbaum A, Hammarling S, McKenney A and Sorensen D (1999) LAPACK Users' Guide (3rd Edition) SIAM, Philadelphia http://www.netlib.org/lapack/lug
Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore
Higham N J (2002) Accuracy and Stability of Numerical Algorithms (2nd Edition) SIAM, Philadelphia

Parameters

Compulsory Input Parameters

1:     uplo – string (length ≥ 1)
If uplo='U', the upper triangle of A is stored.
If uplo='L', the lower triangle of A is stored.
Constraint: uplo='U' or 'L'.
2:     ap: – double array
The dimension of the array ap must be at least max1,n×n+1/2
The n by n symmetric matrix A, packed by columns.
More precisely,
  • if uplo='U', the upper triangle of A must be stored with element Aij in api+jj-1/2 for ij;
  • if uplo='L', the lower triangle of A must be stored with element Aij in api+2n-jj-1/2 for ij.
3:     bldb: – double array
The first dimension of the array b must be at least max1,n.
The second dimension of the array b must be at least max1,nrhs_p.
The n by r right-hand side matrix B.

Optional Input Parameters

1:     n int64int32nag_int scalar
Default: the first dimension of the array b.
n, the number of linear equations, i.e., the order of the matrix A.
Constraint: n0.
2:     nrhs_p int64int32nag_int scalar
Default: the second dimension of the array b.
r, the number of right-hand sides, i.e., the number of columns of the matrix B.
Constraint: nrhs_p0.

Output Parameters

1:     ap: – double array
The dimension of the array ap will be max1,n×n+1/2
The block diagonal matrix D and the multipliers used to obtain the factor U or L from the factorization A=UDUT or A=LDLT as computed by nag_lapack_dsptrf (f07pd), stored as a packed triangular matrix in the same storage format as A.
2:     ipivn int64int32nag_int array
Details of the interchanges and the block structure of D. More precisely,
  • if ipivi=k>0, dii is a 1 by 1 pivot block and the ith row and column of A were interchanged with the kth row and column;
  • if uplo='U' and ipivi-1=ipivi=-l<0, di-1,i-1d-i,i-1 d-i,i-1dii is a 2 by 2 pivot block and the i-1th row and column of A were interchanged with the lth row and column;
  • if uplo='L' and ipivi=ipivi+1=-m<0, diidi+1,idi+1,idi+1,i+1 is a 2 by 2 pivot block and the i+1th row and column of A were interchanged with the mth row and column.
3:     bldb: – double array
The first dimension of the array b will be max1,n.
The second dimension of the array b will be max1,nrhs_p.
If info=0, the n by r solution matrix X.
4:     info int64int32nag_int scalar
info=0 unless the function detects an error (see Error Indicators and Warnings).

Error Indicators and Warnings

Cases prefixed with W are classified as warnings and do not generate an error of type NAG:error_n. See nag_issue_warnings.

   info<0
If info=-i, argument i had an illegal value. An explanatory message is output, and execution of the program is terminated.
W  info>0
Element _ of the diagonal is exactly zero. The factorization has been completed, but the block diagonal matrix D is exactly singular, so the solution could not be computed.

Accuracy

The computed solution for a single right-hand side, x^ , satisfies an equation of the form
A+E x^=b ,  
where
E1 = Oε A1  
and ε  is the machine precision. An approximate error bound for the computed solution is given by
x^-x1 x1 κA E1 A1 ,  
where κA = A-11 A1 , the condition number of A  with respect to the solution of the linear equations. See Section 4.4 of Anderson et al. (1999) and Chapter 11 of Higham (2002) for further details.
nag_lapack_dspsvx (f07pb) is a comprehensive LAPACK driver that returns forward and backward error bounds and an estimate of the condition number. Alternatively, nag_linsys_real_symm_packed_solve (f04bj) solves AX=B  and returns a forward error bound and condition estimate. nag_linsys_real_symm_packed_solve (f04bj) calls nag_lapack_dspsv (f07pa) to solve the equations.

Further Comments

The total number of floating-point operations is approximately 13 n3 + 2n2r , where r  is the number of right-hand sides.
The complex analogues of nag_lapack_dspsv (f07pa) are nag_lapack_zhpsv (f07pn) for Hermitian matrices, and nag_lapack_zspsv (f07qn) for symmetric matrices.

Example

This example solves the equations
Ax=b ,  
where A  is the symmetric matrix
A = -1.81 2.06 0.63 -1.15 2.06 1.15 1.87 4.20 0.63 1.87 -0.21 3.87 -1.15 4.20 3.87 2.07   and   b = 0.96 6.07 8.38 9.50 .  
Details of the factorization of A  are also output.
function f07pa_example


fprintf('f07pa example results\n\n');

% Indefinite matrix A, Upper triangle stored in packed form
uplo = 'Upper';
n = int64(4);
ap = [-1.81  ...
      2.06  1.15   ...
      0.63  1.87  -0.21   ...
     -1.15  4.20   3.87   2.07];

% RHS
b = [0.96;
     6.07;
     8.38;
     9.50];

[apf, ipiv, x, info] = f07pa( ...
                              uplo, ap, b);

disp('Solution');
disp(x');

[ifail] = x04cc( ...
                 uplo, 'Non-unit', n, apf, 'Details of factorization');

fprintf('\nPivot indices\n   ');
fprintf('%11d', ipiv);
fprintf('\n');


f07pa example results

Solution
   -5.0000   -2.0000    1.0000    4.0000

 Details of factorization
             1          2          3          4
 1      0.4074     0.3031    -0.5960     0.6537
 2                -2.5907     0.8115     0.2230
 3                            1.1500     4.2000
 4                                       2.0700

Pivot indices
             1          2         -2         -2

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